• East Asian mathematical journal
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• 제35권1호
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• pp.41-50
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• 2019

This paper is dealt with one-dimensional elliptic jumping problem with nonlinearities crossing n eigenvalues. We get one theorem which shows multiplicity results for solutions of one-dimensional elliptic boundary value problem with jumping nonlinearities. This theorem is that there exist at least two solutions when nonlinearities crossing odd eigenvalues, at least three solutions when nonlinearities crossing even eigenvalues, exactly one solutions and no solution depending on the source term. We obtain these results by the eigenvalues and the corresponding normalized eigenfunctions of the elliptic eigenvalue problem and Leray-Schauder degree theory.

• 호남수학학술지
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• 제26권4호
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• pp.589-608
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• 2004

We investigate the multiplicity of the periodic solutions of the nonlinear wave equation with jumping nonlinearity. By category theory we prove that the jumping problem has at least 2k+1 solutions for the positive source term.

• Korean Journal of Mathematics
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• 제16권4호
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• pp.545-551
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• 2008

We investigate the multiple solutions for a parabolic boundary value problem with jumping nonlinearity crossing no eigenvalues. We show the existence of the unique solution of the parabolic problem with Dirichlet boundary condition and periodic condition when jumping nonlinearity does not cross eigenvalues of the Laplace operator $-{\Delta}$. We prove this result by investigating the Lipschitz constant of the inverse compact operator of $D_t-{\Delta}$ and applying the contraction mapping principle.

• Korean Journal of Mathematics
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• 제26권4호
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• pp.683-700
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• 2018

We get one theorem which shows existence of solutions for one-dimensional jumping problem involving p-Laplacian and Dirichlet boundary condition. This theorem is that there exists at least one solution when nonlinearities crossing finite number of eigenvalues, exactly one solutions and no solution depending on the source term. We obtain these results by the eigenvalues and the corresponding normalized eigenfunctions of the p-Laplacian eigenvalue problem when 1 < p < ${\infty}$, variational reduction method and Leray-Schauder degree theory when $2{\leq}$ p < ${\infty}$.

• 대한수학회지
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• 제37권6호
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• pp.943-956
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• 2000

We investigate a relation between multiplicity of solutions and source terms of jumping problem in wave equation when the nonlinearity crosses an eigenvalue and the source term is generated by finite eigenfunctions. We also show that the jumping problem has infinitely many solutions when the source term is positive multiple of the positve eigenfunction.

• Korean Journal of Mathematics
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• 제19권1호
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• pp.101-109
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• 2011

We get a theorem which shows that there exist at least two or three nontrivial weak solutions for the nonlinear parabolic boundary value problem with the variable coefficient jumping nonlinearity. We prove this theorem by restricting ourselves to the real Hilbert space. We obtain this result by approaching the topological method. We use the Leray-Schauder degree theory on the real Hilbert space.

• Korean Journal of Mathematics
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• 제20권1호
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• pp.125-135
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• 2012

We obtain the multiple solutions for the fourth order elliptic problem with a variable coefficient and a jumping semilinear term. We have a result that there exist at least two solutions if the variable coefficient of the semilinear term crosses some number of the eigenvalues of the biharmonic eigenvalue problem. We obtain this multiplicity result by applying the Leray-Schauder degree theory.

• Korean Journal of Mathematics
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• 제20권4호
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• pp.507-515
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• 2012

We investigate the number of the solutions for the elliptic boundary value problem. We obtain a theorem which shows the existence of six weak solutions for the elliptic problem with jumping nonlinearity crossing three eigenvalues. We get this result by using the geometric mapping defined on the finite dimensional subspace. We use the contraction mapping principle to reduce the problem on the infinite dimensional space to that on the finite dimensional subspace. We construct a three dimensional subspace with three axis spanned by three eigenvalues and a mapping from the finite dimensional subspace to the one dimensional subspace.

• 충청수학회지
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• 제22권2호
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• pp.251-259
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• 2009

We investigate the multiple solutions for the nonlinear parabolic boundary value problem with jumping nonlinearity crossing two eigenvalues. We show the existence of at least four nontrivial periodic solutions for the parabolic boundary value problem. We restrict ourselves to the real Hilbert space and obtain this result by the geometry of the mapping.

• Korean Journal of Mathematics
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• 제17권3호
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• pp.299-314
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• 2009

We investigate the multiple nontrivial solutions of the asymmetric beam equation $u_{tt}+u_{xxxx}=b_1[{(u + 2)}^+-2]+b_2[{(u + 3)}^+-3]$ with Dirichlet boundary condition and periodic condition on t. We reduce this problem into a two-dimensional problem by using variational reduction method and apply the Mountain Pass theorem to find the nontrivial solutions of the equation.